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Eigenvalues and eigenvectors of a Hecke operator on \(S^2\). (Valeurs propres et vecteurs propres d’un opérateur de Hecke sur \(S^2\).) (French) Zbl 0900.11033
Séminaire de théorie spectrale et géométrie. Année 1986-1987. Chambéry: Univ. de Savoie, Fac. des Sciences, Service de Mathématiques, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 5, 165-173 (1987).
The author determines some eigenvalues and eigenvectors associated to the Hecke operator \(T_5\) used by A. Lubotzky, R. Phillips, and P. Sarnak [Commun. Pure Appl. Math. 40, 401-420 (1987; Zbl 0648.10034) and ibid. 39, Suppl., S149–S186 (1986; Zbl 0619.10052)]. \(T_5\) acts stably on \(H_n\), the space of harmonic spheres of degree \(n\). The author demonstrates that, while for \(n\leq 5\) the eigenvalues are necessarily rational, this is not the case for \(n\geq 6\).
For the entire collection see [Zbl 0825.00040].
MSC:
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F55 Other groups and their modular and automorphic forms (several variables)
11P21 Lattice points in specified regions
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